As one would wonder from (REF ), when does the T-E for process level law of Markov chain, increases at worse linearly with dimension (in sample term)?
Sufficient conditions (see e.g., [1]}, [2]}) are
\((i)&\hspace{10.0pt} \mathcal {P}(x,\cdot ) \in T_1^{d}(C), \hspace{72.0pt} \text{for all}~{x \in \mathcal {X}}, \text{and some}~{C>0}, \\(ii)&\hspace{10.0pt} \mathcal {W}_{d}(\mathcal {P}(x,\cdot ),\mathcal {P}(y,\cdot )) \le \hat{\lambda }d(x,y),\hspace{8.0pt} \text{for all }~{(x,y) \in \mathcal {X}^{2}} \text{and some}~{\hat{\lambda } \in [0,1)}.\) 
